One of the reasons the telegrapher's equation has been dismissed as
a candidate for relativistic Brownian motion is that its solution
in space violates Chapman-Kolmogorov's relations.
The standard solution of the telegrapher's equation in References
[1]}, [2]} and
[3]} does not distinguish states with different
direction and uses initial conditions where
\(u_0=1\)  and \(u_0=-1\)  are equally probable.
The solutions discussed by Kac [4]} and Dunkel and Hänggi
[5]} also has mixed initial conditions and is thus a
superposition of two distinct solutions.
